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4
Physical Principles
of RFID Systems
The vast majority of RFID systems operate according to the principle of inductive cou-
pling. Therefore, understanding of the procedures of power and data transfer requires
a thorough grounding in the physical principles of magnetic phenomena. This chapter
therefore contains a particularly intensive study of the theory of magnetic fields from
the point of view of RFID.
Electromagnetic fields — radio waves in the classic sense — are used in RFID
systems that operate at above 30 MHz. To aid understanding of these systems we
will investigate the propagation of waves in the far field and the principles of radar
technology.
Electric fields play a secondary role and are only exploited for capacitive data
transmission in close coupling systems. Therefore, this type of field will not be dis-
cussed further.
4.1 Magnetic Field
4.1.1 Magnetic field strength
H
Every moving charge (electrons in wires or in a vacuum), i.e. flow of current, is
associated with a magnetic field (Figure 4.1). The intensity of the magnetic field can
be demonstrated experimentally by the forces acting on a magnetic needle (compass)
or a second electric current. The magnitude of the magnetic field is described by the
magnetic field strength H regardless of the material properties of the space.
In the general form we can say that: ‘the contour integral of magnetic field strength
along a closed curve is equal to the sum of the current strengths of the currents within
it’ (Kuchling, 1985).

I =


H ·


ds(4.1)
We can use this formula to calculate the field strength H for different types of
conductor. See Figure 4.2.
RFID Handbook: Fundamentals and Applications in Contactless Smart Cards and Identification,
Second Edition
Klaus Finkenzeller
Copyright
 2003 John Wiley & Sons, Ltd.
ISBN: 0-470-84402-7
62 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
I
+
−
Magnetic flux
lines
Figure 4.1 Lines of magnetic flux are generated around every current-carrying conductor
I
H
I
+
−
−
+
H
Figure 4.2 Lines of magnetic flux around a current-carrying conductor and a current-carrying
cylindrical coil
Table 4.1 Constants used
Constant Symbol Value and unit
Electric field constant ε

0
8.85 ×10
−12
As/Vm
Magnetic field constant µ
0
1.257 ×10
−6
Vs/Am
Speed of light c 299 792 km/s
Boltzmann constant k 1.380 662 × 10
−23
J/K
In a straight conductor the field strength H along a circular flux line at a distance
r is constant. The following is true (Kuchling, 1985):
H =
1
2πr
(4.2)
4.1.1.1 Path of field strength H
(
x
)
in conductor loops
So-called ‘short cylindrical coils’ or conductor loops are used as magnetic antennas to
generate the magnetic alternating field in the write/read devices of inductively coupled
RFID systems (Figure 4.3).
4.1 MAGNETIC FIELD 63
Table 4.2 Units and abbreviations used
Variable Symbol Unit Abbreviation

Magnetic field strength H Ampere per meter A/m
Magnetic flux (n = number
of windings)
 Volt seconds Vs
 = n
Magnetic inductance B Volt seconds per meter
squared
Vs/m
2
Inductance L Henry H
Mutual inductance M Henry H
Electric field strength E Volts per metre V/m
Electric current I Ampere A
Electric voltage U Vo l t V
Capacitance C Farad F
Frequency f Hertz Hz
Angular frequency ω = 2πf 1/seconds 1/s
Length l Metre m
Area A Metre squared m
2
Speed v Metres per second m/s
Impedance Z Ohm 
Wavelength λ Metre m
Power P Watt W
Power density S Watts per metre squared W/m
2
d
r
H
x

Figure 4.3 The path of the lines of magnetic flux around a short cylindrical coil, or conductor
loop, similar to those employed in the transmitter antennas of inductively coupled RFID systems
If the measuring point is moved away from the centre of the coil along the coil axis
(x axis), then the strength of the field H will decrease as the distance x is increased.
A more in-depth investigation shows that the field strength in relation to the radius
(or area) of the coil remains constant up to a certain distance and then falls rapidly
(see Figure 4.4). In free space, the decay of field strength is approximately 60 dB per
64 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
0.01
0.1
10
100
Magnetic field strength
H
(A/m)
1×10
−3
1×10
−4
1×10
−5
1×10
−6
1×10
−7
1×10
−8
0.01 0.1 1 10
R
= 55 cm

R
= 7.5 cm
R
= 1 cm
Distance
x
(m)
1×10
−3
Figure 4.4 Path of magnetic field strength H in the near field of short cylinder coils, or
conductor coils, as the distance in the x direction is increased
decade in the near field of the coil, and flattens out to 20 dB per decade in the far field
of the electromagnetic wave that is generated (a more precise explanation of these
effects can be found in Section 4.2.1).
The following equation can be used to calculate the path of field strength along the
x axis of a round coil (= conductor loop) similar to those employed in the transmitter
antennas of inductively coupled RFID systems (Paul, 1993):
H =
I · N ·R
2
2

(R
2
+ x
2
)
3
(4.3)
where N is the number of windings, R is the circle radius r and x is the distance from

the centre of the coil in the x direction. The following boundary condition applies to
this equation: d  R and x<λ/2π (the transition into the electromagnetic far field
begins at a distance >2π; see Section 4.2.1).
At distance 0 or, in other words, at the centre of the antenna, the formula can be
simplified to (Kuchling, 1985):
H =
I · N
2R
(4.4)
We can calculate the field strength path of a rectangular conductor loop with edge
length a ×b at a distance of x using the following equation. This format is often used
4.1 MAGNETIC FIELD 65
as a transmitter antenna.
H =
N ·I ·ab
4π


a
2

2
+

b
2

2
+ x
2

·





1

a
2

2
+ x
2
+
1

b
2

2
+ x
2





(4.5)
Figure 4.4 shows the calculated field strength path H(x) for three different antennas

at a distance 0–20 m. The number of windings and the antenna current are constant
in each case; the antennas differ only in radius R. The calculation is based upon the
following values: H 1: R = 55 cm, H 2: R = 7.5cm, H 3: R = 1cm.
The calculation results confirm that the increase in field strength flattens out at short
distances (x<R) from the antenna coil. Interestingly, the smallest antenna exhibits
a significantly higher field strength at the centre of the antenna (distance = 0), but
at greater distances (x>R) the largest antenna generates a significantly higher field
strength. It is vital that this effect is taken into account in the design of antennas for
inductively coupled RFID systems.
4.1.1.2 Optimal antenna diameter
If the radius R of the transmitter antenna is varied at a constant distance x from the
transmitter antenna under the simplifying assumption of constant coil current I in the
transmitter antenna, then field strength H is found to be at its highest at a certain ratio
of distance x to antenna radius R. This means that for every read range of an RFID
system there is an optimal antenna radius R. This is quickly illustrated by a glance at
Figure 4.4: if the selected antenna radius is too great, the field strength is too low even
at a distance x = 0 from the transmission antenna. If, on the other hand, the selected
antenna radius is too small, then we find ourselves within the range in which the field
strength falls in proportion to x
3
.
Figure 4.5 shows the graph of field strength H as the coil radius R is varied.
The optimal coil radius for different read ranges is always the maximum point of
the graph H(R). To find the mathematical relationship between the maximum field
strength H and the coil radius R we must first find the inflection point of the function
H(R) (see equation 4.3) (Lee, 1999). To do this we find the first derivative H

(R) by
differentiating H(R) with respect to R:
H


(R) =
d
dR
H(R) =
2 ·I ·N ·R

(R
2
+ x
2
)
3
−
3 ·I ·N ·R
3
(R
2
+ x
2
) ·

(R
2
+ x
2
)
3
(4.6)
The inflection point, and thus the maximum value of the function H(R), is found

from the following zero points of the derivative H

(R):
R
1
= x ·
√
2; R
2
=−x ·
√
2 (4.7)
66 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
Radius
R
(m)
Magnetic field strength
H
(A/m)
0
1
2
3
4
0 0.2 0.4 0.6 0.8 1
x
= 10 cm
x
= 20 cm
x

= 30 cm
1.5 A/m (ISO 14443)
Figure 4.5 Field strength H of a transmission antenna given a constant distance x and variable
radius R,whereI = 1AandN = 1
The optimal radius of a transmission antenna is thus twice the maximum desired
read range. The second zero point is negative merely because the magnetic field H of
a conductor loop propagates in both directions of the x axis (see also Figure 4.3).
However, an accurate assessment of a system’s maximum read range requires
knowledge of the interrogation field strength H
min
of the transponder in question (see
Section 4.1.9). If the selected antenna radius is too great, then there is the danger that
the field strength H may be too low to supply the transponder with sufficient operating
energy, even at a distance x = 0.
4.1.2 Magnetic flux and magnetic flux density
The magnetic field of a (cylindrical) coil will exert a force on a magnetic needle.
If a soft iron core is inserted into a (cylindrical) coil — all other things remaining
equal — then the force acting on the magnetic needle will increase. The quotient
I × N (Section 4.1.1) remains constant and therefore so does field strength. However,
the flux density — the total number of flux lines — which is decisive for the force
generated (cf. Pauls, 1993), has increased.
The total number of lines of magnetic flux that pass through the inside of a cylin-
drical coil, for example, is denoted by magnetic flux . Magnetic flux density B is
a further variable related to area A (this variable is often referred to as ‘magnetic
inductance B in the literature’) (Reichel, 1980). Magnetic flux is expressed as:
 = B ·A(4.8)
4.1 MAGNETIC FIELD 67
Magnetic flux Φ
Area
A

B
line
Figure 4.6 Relationship between magnetic flux  and flux density B
The material relationship between flux density B and field strength H (Figure 4.6)
is expressed by the material equation:
B = µ
0
µ
r
H = µH(4.9)
The constant µ
0
is the magnetic field constant (µ
0
= 4π ×10
−6
Vs/Am) and
describes the permeability (= magnetic conductivity) of a vacuum. The variable µ
r
is called relative permeability and indicates how much greater than or less than µ
0
the
permeability of a material is.
4.1.3 Inductance
L
A magnetic field, and thus a magnetic flux , will be generated around a conductor
of any shape. This will be particularly intense if the conductor is in the form of a loop
(coil). Normally, there is not one conduction loop, but N loops of the same area A,
through which the same current I flows. Each of the conduction loops contributes the
same proportion  to the total flux ψ (Paul, 1993).

 =

N

N
= N · = N ·µ ·H · A(4.10)
The ratio of the interlinked flux ψ that arises in an area enclosed by current I ,to
the current in the conductor that encloses it (conductor loop) is denoted by inductance
L (Figure 4.7):
L =

I
=
N ·
I
=
N ·µ ·H · A
I
(4.11)
I
Φ
(
I
)
, Ψ
(
I
)
Enclosing
current

A
Figure 4.7 Definition of inductance L
68 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
Inductance is one of the characteristic variables of conductor loops (coils). The
inductance of a conductor loop (coil) depends totally upon the material properties
(permeability) of the space that the flux flows through and the geometry of the layout.
4.1.3.1 Inductance of a conductor loop
If we assume that the diameter d of the wire used is very small compared to the
diameter D of the conductor coil (d/D < 0.0001) a very simple approximation can
be used:
L = N
2
µ
0
R ·ln

2R
d

(4.12)
where R is the radius of the conductor loop and d is the diameter of the wire used.
4.1.4 Mutual inductance
M
If a second conductor loop 2 (area A
2
) is located in the vicinity of conductor loop 1
(area A
1
), through which a current is flowing, then this will be subject to a proportion of
the total magnetic flux  flowing through A

1
. The two circuits are connected together
by this partial flux or coupling flux. The magnitude of the coupling flux ψ
21
depends
upon the geometric dimensions of both conductor loops, the position of the conductor
loops in relation to one another, and the magnetic properties of the medium (e.g.
permeability) in the layout.
Similarly to the definition of the (self) inductance L of a conductor loop, the mutual
inductance M
21
of conductor loop 2 in relation to conductor loop 1 is defined as the
ratio of the partial flux ψ
21
enclosed by conductor loop 2, to the current I
1
in conductor
loop 1 (Paul, 1993):
M
21
=

21
(I
1
)
I
1
=


A2
B
2
(I
1
)
I
1
· dA
2
(4.13)
Similarly, there is also a mutual inductance M
12
. Here, current I
2
flows through the
conductor loop 2, thereby determining the coupling flux ψ
12
in loop 1. The following
relationship applies:
M = M
12
= M
21
(4.14)
Mutual inductance describes the coupling of two circuits via the medium of a mag-
netic field (Figure 4.8). Mutual inductance is always present between two electric
circuits. Its dimension and unit are the same as for inductance.
The coupling of two electric circuits via the magnetic field is the physical prin-
ciple upon which inductively coupled RFID systems are based. Figure 4.9 shows a

calculation of the mutual inductance between a transponder antenna and three dif-
ferent reader antennas, which differ only in diameter. The calculation is based upon
the following values: M
1
: R = 55 cm, M
2
: R = 7.5cm, M
3
: R = 1 cm, transponder:
R = 3.5cm. N = 1 for all reader antennas.
The graph of mutual inductance shows a strong similarity to the graph of magnetic
field strength H along the x axis. Assuming a homogeneous magnetic field, the mutual
4.1 MAGNETIC FIELD 69
Φ(
I
1
), Ψ(
I
1
)
B
2
(
I
1
)
I
1
Total flux
Ψ

2
(
I
1
)
A
2
A
1
Figure 4.8 The definition of mutual inductance M
21
by the coupling of two coils via a partial
magnetic flow
0.01
Distance
x
(m)
1×10
−3
0.1 1
1×10
−7
1×10
−8
1×10
−9
1×10
−10
Mutual inductance
M

(Henry)
1×10
−11
1×10
−12
1×10
−13
1×10
−14
M1
M2
M3
Figure 4.9 Graph of mutual inductance between reader and transponder antenna as the distance
in the x direction increases
inductance M
12
between two coils can be calculated using equation (4.13). It is found
to be:
M
12
=
B
2
(I
1
) ·N
2
· A
2
I

1
=
µ
0
· H(I
1
) ·N
2
· A
2
I
1
(4.15)
We first replace H(I
1
) with the expression in equation (4.4), and substitute R
2
π for
A, thus obtaining:
M
12
=
µ
0
· N
1
· R
2
1
· N

2
· R
2
2
· π
2

(R
2
1
+ x
2
)
3
(4.16)
70 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
In order to guarantee the homogeneity of the magnetic field in the area A
2
the
condition A
2
≤ A
1
should be fulfilled. Furthermore, this equation only applies to the
case where the x axes of the two coils lie on the same plane. Due to the relationship
M = M
12
= M
21
the mutual inductance can be calculated as follows for the case

A
2
≥ A
1
:
M
21
=
µ
0
· N
1
· R
2
1
· N
2
· R
2
2
· π
2

(R
2
2
+ x
2
)
3

(4.17)
4.1.5 Coupling coefficient
k
Mutual inductance is a quantitative description of the flux coupling of two conductor
loops. The coupling coefficient k is introduced so that we can make a qualitative
prediction about the coupling of the conductor loops independent of their geometric
dimensions. The following applies:
k =
M
√
L
1
· L
2
(4.18)
The coupling coefficient always varies between the two extreme cases 0 ≤ k ≤ 1.
• k = 0: Full decoupling due to great distance or magnetic shielding.
• k = 1: Total coupling. Both coils are subject to the same magnetic flux .The
transformer is a technical application of total coupling, whereby two or more coils
are wound onto a highly permeable iron core.
An analytic calculation is only possible for very simple antenna configurations.
For two parallel conductor loops centred on a single x axis the coupling coefficient
according to Roz and Fuentes (n.d.) can be approximated from the following equation.
However, this only applies if the radii of the conductor loops fulfil the condition
r
Transp
≤ r
Reader
. The distance between the conductor loops on the x axis is denoted
by x.

k(x) ≈
r
2
Transp
· r
2
Reader
√
r
Transp
· r
Reader
·


x
2
+ r
2
Reader

3
(4.19)
Due to the fixed link between the coupling coefficient and mutual inductance M,
and because of the relationship M = M
12
= M
21
, the formula is also applicable to
transmitter antennas that are smaller than the transponder antenna. Where r

Transp
≥
r
Reader
, we write:
k(x) ≈
r
2
Transp
· r
2
Reader
√
r
Transp
· r
Reader
·


x
2
+ r
2
Transp

3
(4.20)
The coupling coefficient k(x) = 1(= 100%) is achieved where the distance between
the conductor loops is zero (x = 0) and the antenna radii are identical (r

Transp
= r
Reader
),
4.1 MAGNETIC FIELD 71
Coupling coefficient
k
=
f
(
x
)
Distance
x
(m)
Coupling coefficient
k
(
x
)
0.05
0.1
0.15
0.2
0.25
0.011× 10
−3
0.1 1
r
1

r
2
r
3
Figure 4.10 Graph of the coupling coefficient for different sized conductor loops. Transponder
antenna: r
Transp
= 2 cm, reader antenna: r
1
= 10 cm, r
2
= 7.5cm, r
3
= 1cm
because in this case the conductor loops are in the same place and are exposed to exactly
the same magnetic flux ψ.
In practice, however, inductively coupled transponder systems operate with coupling
coefficients that may be as low as 0.01 (<1%) (Figure 4.10).
4.1.6 Faraday’s law
Any change to the magnetic flux  generates an electric field strength E
i
. This char-
acteristic of the magnetic field is described by Faraday’s law.
The effect of the electric field generated in this manner depends upon the material
properties of the surrounding area. Figure 4.11 shows some of the possible effects
(Paul, 1993):
• Vacuum: in this case, the field strength E gives rise to an electric rotational field.
Periodic changes in magnetic flux (high frequency current in an antenna coil)
generate an electromagnetic field that propagates itself into the distance.
• Open conductor loop: an open circuit voltage builds up across the ends of an almost

closed conductor loop, which is normally called induced voltage. This voltage
corresponds with the line integral (path integral) of the field strength E that is
generated along the path of the conductor loop in space.
72 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
U
i
Flux change dΦ/d
t
Conductor (e.g. metal surface)
Eddy current,
Current density
S
Open conductor loop
Nonconductor (vacuum),
Induced field strength
E
i
=> Electromagnetic wave
Figure 4.11 Induced electric field strength E in different materials. From top to bottom: metal
surface, conductor loop and vacuum
• Metal surface: an electric field strength E is also induced in the metal surface. This
causes free charge carriers to flow in the direction of the electric field strength.
Currents flowing in circles are created, so-called eddy currents. This works against
the exciting magnetic flux (Lenz’s law), which may significantly damp the mag-
netic flux in the vicinity of metal surfaces. However, this effect is undesirable in
inductively coupled RFID systems (installation of a transponder or reader antenna
on a metal surface) and must therefore be prevented by suitable countermeasures
(see Section 4.1.12.3).
In its general form Faraday’s law is written as follows:
u

i
=

E
i
· ds =−
d(t)
dt
(4.21)
For a conductor loop configuration with N windings, we can also say that u
i
= N ·
d/dt. (The value of the contour integral
∫
E
i
· ds can be increased N times if the
closed integration path is carried out N times; Paul, 1993).
To improve our understanding of inductively coupled RFID systems we will now
consider the effect of inductance on magnetically coupled conduction loops.
A time variant current i
1
(t) in conduction loop L
1
generates a time variant magnetic
flux d(i
1
)/dt. In accordance with the inductance law, a voltage is induced in the
conductor loops L
1

and L
2
through which some degree of magnetic flux is flowing.
We can differentiate between two cases:
• Self-inductance: the flux change generated by the current change di
n
/dt induces a
voltage u
n
in the same conductor circuit.
• Mutual inductance: the flux change generated by the current change di
n
/dt induces
a voltage in the adjacent conductor circuit L
m
. Both circuits are coupled by
mutual inductance.
Figure 4.12 shows the equivalent circuit diagram for coupled conductor loops. In an
inductively coupled RFID system L
1
would be the transmitter antenna of the reader.
4.1 MAGNETIC FIELD 73
u
1
u
2
i
1
i
2

M
M
u
2
R
2
R
L
L
1
L
2
B
2
(
i
1
)
L
1
L
2
Figure 4.12 Left, magnetically coupled conductor loops; right, equivalent circuit diagram for
magnetically coupled conductor loops
L
2
represents the antenna of the transponder, where R
2
is the coil resistance of the
transponder antenna. The current consumption of the data memory is symbolised by

the load resistor R
L
.
A time varying flux in the conductor loop L
1
induces voltage u
2i
in the conductor
loop L
2
due to mutual inductance M. The flow of current creates an additional voltage
drop across the coil resistance R
2
, meaning that the voltage u
2
can be measured at the
terminals. The current through the load resistor R
L
is calculated from the expression
u
2
/R
L
. The current through L
2
also generates an additional magnetic flux, which
opposes the magnetic flux 
1
(i
1

). The above is summed up in the following equation:
u
2
=+
d
2
dt
= M
di
1
dt
− L
2
di
2
dt
− i
2
R
2
(4.22)
Because, in practice, i
1
and i
2
are sinusoidal (HF) alternating currents, we write
equation (4.22) in the more appropriate complex notation (where ω = 2πf):
u
2
= jωM ·i

1
− jωL
2
· i
2
− i
2
R
2
(4.23)
If i
2
is replaced by u
2
/R
L
in equation (4.23), then we can solve the equation for u
2
:
u
2
=
jwM · i
1
1 +
jwL
2
+ R
2
R

L
R
L
→ ∞: u
2
= jωM · i
1
R
L
→ 0: u
2
→ 0
(4.24)
4.1.7 Resonance
The voltage u
2
induced in the transponder coil is used to provide the power supply to
thedatamemory(microchip) of a passive transponder (see Section 4.1.8.1). In order to
significantly improve the efficiency of the equivalent circuit illustrated in Figure 4.12,
an additional capacitor C
2
is connected in parallel with the transponder coil L
2
to
form a parallel resonant circuit with a resonant frequency that corresponds with the
74 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
operating frequency of the RFID system in question.
1
The resonant frequency of the
parallel resonant circuit can be calculated using the Thomson equation:

f =
1
2π
√
L
2
· C
2
(4.25)
In practice, C
2
is made up of a parallel capacitor C

2
and a parasitic capacitance
C
p
from the real circuit. C
2
= (C

2
+ C
p
). The required capacitance for the parallel
capacitor C

2
is found using the Thomson equation, taking into account the parasitic
capacitance C

p
:
C

2
=
1
(2πf )
2
L
2
− C
p
(4.26)
Figure 4.13 shows the equivalent circuit diagram of a real transponder. R
2
is the
natural resistance of the transponder coil L
2
and the current consumption of the data
carrier (chip) is represented by the load resistor R
L
.
If a voltage u
Q2
= u
i
is induced in the coil L
2
, the following voltage u

2
can be
measured at the data carrier load resistor R
L
in the equivalent circuit diagram shown
in Figure 4.13:
u
2
=
u
Q2
1 +(jωL
2
+ R
2
) ·

1
R
L
+ jωC
2

(4.27)
We now replace the induced voltage u
Q2
= u
i
by the factor responsible for its
generation, u

Q2
= u
i
= jωM ·i
1
= ω ·k ·
√
L
1
· L
2
· i
1
, thus obtaining the relationship
u
Q2
C
p
i
1
∼
i
2
M
u
2
R
2
R
L

L
1
L
2
C
′
2
C
2
=
C
p
+
C
′
2
Parasitic capacitor
Parallel C (‘tuning C’)
Figure 4.13 Equivalent circuit diagram for magnetically coupled conductor loops. Transponder
coil L
2
and parallel capacitor C
2
form a parallel resonant circuit to improve the efficiency of
voltage transfer. The transponder’s data carrier is represented by the grey box
1
However, in 13.56 MHz systems with anticollision procedures, the resonant frequency selected for the
transponder is often 1–5 MHz higher to minimise the effect of the interaction between transponders on
overall performance. This is because the overall resonant frequency of two transponders directly adjacent
to one another is always lower than the resonant frequency of a single transponder.

4.1 MAGNETIC FIELD 75
between voltage u
2
and the magnetic coupling of transmitter coil and transponder coil:
u
2
=
jωM · i
1
1 +(jωL
2
+ R
2
) ·

1
R
L
+ jωC
2

(4.28)
and:
u
2
=
jω ·k ·
√
L
1

· L
2
· i
1
1 +(jωL
2
+ R
2
) ·

1
R
L
+ jωC
2

(4.29)
or in the non-complex form (Jurisch, 1994):
u
2
=
ω · k ·
√
L
1
L
2
· i
1



ωL
2
R
L
+ ωR
2
C
2

2
+

1 −ω
2
L
2
C
2
+
R
2
R
L

2
(4.30)
where C
2
= C


2
+ C
p
.
Figure 4.14 shows the simulated graph of u
2
with and without resonance over a large
frequency range for a possible transponder system. The current i
1
in the transmitter
antenna (and thus also (i
1
)), inductance L
2
, mutual inductance M, R
2
and R
L
are
held constant over the entire frequency range.
We see that the graph of voltage u
2
for the circuit with the coil alone (circuit from
Figure 4.12) is almost identical to that of the parallel resonant circuit (circuit from
0.1
1
10
100
|

u
2
| (V)
1 × 10
6
1 × 10
7
1 × 10
8
u
2
Resonant
u
2
Coil
f
(Hz)
Figure 4.14 Plot of voltage at a transponder coil in the frequency range 1 to 100 MHz,
given a constant magnetic field strength H or constant current i
1
. A transponder coil with
a parallel capacitor shows a clear voltage step-up when excited at its resonant frequency
(f
RES
= 13.56 MHz)
76 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
Figure 4.13) at frequencies well below the resonant frequencies of both circuits, but
that when the resonant frequency is reached, voltage u
2
increases by more than a

power of ten in the parallel resonant circuit compared to the voltage u
2
for the coil
alone. Above the resonant frequency, however, voltage u
2
falls rapidly in the parallel
resonant circuit, even falling below the value for the coil alone.
For transponders in the frequency range below 135 kHz, the transponder coil L
2
is
generally connected in parallel with a chip capacitor (C

2
= 20–220 pF) to achieve the
desired resonant frequency. At the higher frequencies of 13.56 MHz and 27.125 MHz,
the required capacitance C
2
is usually so low that it is provided by the input capacitance
of the data carrier together with the parasitic capacitance of the transponder coil.
Let us now investigate the influence of the circuit elements R
2
, R
L
and L
2
on
voltage u
2
. To gain a better understanding of the interactions between the individual
parameters we will now introduce the Q factor (the Q factor crops up again when we

investigate the connection of transmitter antennas in Section 11.4.1.3). We will refrain
from deriving formulas because the electric resonant circuit is dealt with in detail in
the background reading.
The Qfactor is a measure of the voltage and current step-up in the resonant circuit
at its resonant frequency. Its reciprocal 1/Q denotes the expressively named circuit
damping d. The Q factor is very simple to calculate for the equivalent circuit in
Figure 4.13. In this case ω is the angular frequency (ω = 2πf ) of the transponder
resonant circuit:
Q =
1
R
2
·

C
2
L
2
+
1
R
L
·

L
2
C
2
=
1

R
2
ωL
2
+
ωL
2
R
L
(4.31)
A glance at equation (4.31) shows that when R
2
→∞and R
L
→ 0, the Q factor
also tends towards zero. On the other hand, when the transponder coil has a very low
coil resistance R
2
→ 0 and there is a high load resistor R
L
 0 (corresponding with
very low transponder chip power consumption), very high Q factors can be achieved.
The voltage u
2
is now proportional to the quality of the resonant circuit, which means
that the dependency of voltage u
2
upon R
2
and R

L
is clearly defined.
Voltage u
2
thus tends towards zero where R
2
→∞and R
L
→ 0. At a very low
transponder coil resistance R
2
→ 0 and a high value load resistor R
L
 0, on the other
hand, a very high voltage u
2
can be achieved (compare equation (4.30)).
It is interesting to note the path taken by the graph of voltage u
2
when the inductance
of the transponder coil L
2
is changed, thus maintaining the resonance condition (i.e.
C
2
= 1/ω
2
L
2
for all values of L

2
). We see that for certain values of L
2
, voltage u
2
reaches a clear peak (Figure 4.15).
If we now consider the graph of the Q factor as a function of L
2
(Figure 4.16), then
we observe a maximum at the same value of transponder inductance L
2
. The maximum
voltage u
2
= f(L
2
) is therefore derived from the maximum Q factor, Q = f(L
2
),at
this point.
This indicates that for every pair of parameters (R
2
, R
L
), there is an inductance
value L
2
at which the Q factor, and thus also the supply voltage u
2
to the data carrier,

is at a maximum. This should always be taken into consideration when designing
a transponder, because this effect can be exploited to optimise the energy range of
4.1 MAGNETIC FIELD 77
0
10
20
30
40
k
= 3%,
R
L
= 2
k
,
L
1
= 1 µH
k
= 3%,
R
L
= 2
k
,
L
1
= 2 µH
k
= 5%,

R
L
= 2
k
,
L
1
= 1 µH
1 × 10
−6
1 × 10
−7
L
2
(H)
1 × 10
−5
|
u
2
(
L
2
)| (V)
Figure 4.15 Plot of voltage u
2
for different values of transponder inductance L
2
. The resonant
frequency of the transponder is equal to the transmission frequency of the reader for all values

of L
2
(i
1
= 0.5A, f = 13.56 MHz, R
2
= 1 )
0
5
10
15
20
25
L
(H)
1 × 10
−7
1 × 10
−6
1 × 10
−5
Q factor
Q

=
f
(
L
2
)

Figure 4.16 Graph of the Q factor as a function of transponder inductance L
2
,wherethe
resonant frequency of the transponder is constant (f = 13.56 MHz, R
2
= 1 )
78 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
an inductively coupled RFID system. However, we must also bear in mind that the
influence of component tolerances in the system also reaches a maximum in the Q
max
range. This is particularly important in systems designed for mass production. Such
systems should be designed so that reliable operation is still possible in the range
Q  Q
max
at the maximum distance between transponder and reader.
R
L
should be set at the same value as the input resistance of the data carrier after
setting the ‘power on’ reset, i.e. before the activation of the voltage regulator, as is the
case for the maximum energy range of the system.
4.1.8 Practical operation of the transponder
4.1.8.1 Power supply to the transponder
Transponders are classified as active or passive depending upon the type of power
supply they use.
Active transponders incorporate their own battery to provide the power supply to the
data carrier. In these transponders, the voltage u
2
is generally only required to generate
a ‘wake up’ signal. As soon as the voltage u
2

exceeds a certain limit this signal is
activated and puts the data carrier into operating mode. The transponder returns to the
power saving ‘sleep’ or ‘stand-by mode’ after the completion of a transaction with the
reader, or when the voltage u
2
falls below a minimum value.
In passive transponders the data carrier has to obtain its power supply from the
voltage u
2
. To achieve this, the voltage u
2
is converted into direct current using a low
loss bridge rectifier and then smoothed. A simple basic circuit for this application is
shown in Figure 3.18.
4.1.8.2 Voltage regulation
The induced voltage u
2
in the transponder coil very rapidly reaches high values due to
resonance step-up in the resonant circuit. Considering the example in Figure 4.14, if
we increase the coupling coefficient k — possibly by reducing the gap between reader
and transponder — or the value of the load resistor R
L
, then voltage u
2
will reach
a level much greater than 100 V. However, the operation of a data carrier requires a
constant operating voltage of 3–5 V (after rectification).
In order to regulate voltage u
2
independently of the coupling coefficient k or other

parameters, and to hold it constant in practice, a voltage-dependent shunt resistor R
S
is connected in parallel with the load resistor R
L
. The equivalent circuit diagram for
this is shown in Figure 4.17.
R
L
i
1
i
2
M
u
2
R
2
L
1
L
2
C
2
R
s
Figure 4.17 Operating principle for voltage regulation in the transponder using a shunt
regulator
4.1 MAGNETIC FIELD 79
As induced voltage u
Q2

= u
i
increases, the value of the shunt resistor R
S
falls,
thus reducing the quality of the transponder resonant circuit to such a degree that the
voltage u
2
remains constant. To calculate the value of the shunt resistor for differ-
ent variables, we refer back to equation (4.29) and introduce the parallel connection
of R
L
and R
S
in place of the constant load resistor R
L
. The equation can now be
solved with respect to R
S
. The variable voltage u
2
is replaced by the constant voltage
u
Transp
— the desired input voltage of the data carrier — giving the following equation
for R
S
:
R
S

=








1

jω ·k ·
√
L
1
L
2
· i
1
u
Transp

− 1
jωL
2
+ R
2
− jωC
2
−

1
R
L








|u
2-unreg
>u
Transp
(4.32)
Figure 4.18 shows the graph of voltage u
2
when such an ‘ideal’ shunt regulator is
used. Voltage u
2
initially increases in proportion with the coupling coefficient k.When
u
2
reaches its desired value, the value of the shunt resistor begins to fall in inverse
proportion to k, thus maintaining an almost constant value for voltage u
2
.
Figure 4.19 shows the variable value of the shunt resistor R
S

as a function of the
coupling coefficient. In this example the value range for the shunt resistor covers several
powers of ten. This can only be achieved using a semiconductor circuit, therefore
so-called shunt or parallel regulators are used in inductively coupled transponders.
These terms describe an electronic regulator circuit, the internal resistance of which
0 0.05 0.1 0.15 0.2 0.25 0.3
0
5
10
15
20
u
2
regulated
u
2
unregulated
k
u
2
(V)
Figure 4.18 Example of the path of voltage u
2
with and without shunt regulation in the
transponder, where the coupling coefficient k is varied by altering the distance between transpon-
der and reader antenna. (The calculation is based upon the following parameters: i
1
= 0.5A,
L
1

= 1 µH, L
2
= 3.5 µH, R
L
= 2k, C
2
= 1/ω
2
L
2
)
80 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
0 0.05 0.1 0.15 0.2 0.25 0.3
10
100
k
1 × 10
4
1 × 10
5
1 × 10
3
R
shunt
(Ohm)
R
shunt
=
f
(

k
)
Figure 4.19 The value of the shunt resistor R
S
must be adjustable over a wide range to keep
voltage u
2
constant regardless of the coupling coefficient k (parameters as Figure 4.18)
ZD 4.2
Chip
u
2
V
cc
Rectifier
Figure 4.20 Example circuit for a simple shunt regulator
falls disproportionately sharply when a threshold voltage is exceeded. A simple shunt
regulator based upon a zener diode (N
¨
uhrmann, 1994) is shown in Figure 4.20.
4.1.9 Interrogation field strength
H
min
We can now use the results obtained in Section 4.1.7 to calculate the interrogation
field strength of a transponder. This is the minimum field strength H
min
(at a maximum
distance x between transponder and reader) at which the supply voltage u
2
is just high

enough for the operation of the data carrier.
However, u
2
is not the internal operating voltage of the data carrier (3V or 5V)
here; it is the HF input voltage at the terminal of the transponder coil L
2
on the
data carrier, i.e. prior to rectification. The voltage regulator (shunt regulator) should
not yet be active at this supply voltage. R
L
corresponds with the input resistance of
the data carrier after the ‘power on reset’, C
2
is made up of the input capacitance
4.1 MAGNETIC FIELD 81
C
p
of the data carrier (chip) and the parasitic capacitance of the transponder layout
C

2
: C
2
= (C

2
+ C
p
).
The inductive voltage (source voltage u

Q2
= u
i
) of a transponder coil can be cal-
culated using equation (4.21) for the general case. If we assume a homogeneous,
sinusoidal magnetic field in air (permeability constant = µ
0
) we can derive the fol-
lowing, more appropriate, formula:
u
i
= µ
0
· A ·N ·ω ·H
eff
(4.33)
where H
eff
is the effective field strength of a sinusoidal magnetic field, ω is the angular
frequency of the magnetic field, N is the number of windings of the transponder coil
L
2
,andA is the cross-sectional area of the transponder coil.
We now replace u
Q2
= u
i
= jωM ·i
1
from equation (4.29) with equation (4.33) and

thus obtain the following equation for the circuit in Figure 4.13:
u
2
=
jω ·µ
0
· H
eff
· A ·N
1 +(jωL
2
+ R
2
)

1
R
L
+ jωC
2

(4.34)
Multiplying out the denominator:
u
2
=
jω ·µ
0
· H
eff

· A ·N
jω

L
2
R
L
+ R
2
C
2

+

1 −ω
2
L
2
C
2
+
R
2
R
L

(4.35)
We now solve this equation for H
eff
and obtain the value of the complex form. This

yields the following relationship for the interrogation field H
min
in the general case:
H
min
=
u
2
·


ωL
2
R
L
+ ωR
2
C
2

2
+

1 −ω
2
L
2
C
2
+

R
2
R
L

2
ω · µ
0
· A ·N
(4.36)
A more detailed analysis of equation (4.36) shows that the interrogation field strength
is dependent upon the frequency ω = 2πf in addition to the antenna area A, the num-
ber of windings N (of the transponder coil), the minimum voltage u
2
and the input
resistance R
2
. This is not surprising, because we have determined a resonance step-up
of u
2
at the resonant frequency of the transponder resonant circuit. Therefore, when
the transmission frequency of the reader corresponds with the resonant frequency of
the transponder, the interrogation field strength H
min
is at its minimum value.
To optimise the interrogation sensitivity of an inductively coupled RFID system, the
resonant frequency of the transponder should be matched precisely to the transmission
frequency of the reader. Unfortunately, this is not always possible in practice. First,
tolerances occur during the manufacture of a transponder, which lead to a deviation
in the transponder resonant frequency. Second, there are also technical reasons for

setting the resonant frequency of the transponder a few percentage points higher than
the transmission frequency of the reader (for example in systems using anticollision
procedures to keep the interaction of nearby transponders low).
82 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
Some semiconductor manufacturers incorporate additional smoothing capacitors into
the transponder chip to smooth out frequency deviations in the transponder caused
by manufacturing tolerances (see Figure 3.28, ‘tuning C’). During manufacture the
transponder is adjusted to the desired frequency by switching individual smoothing
capacitors on and off (Sch
¨
urmann, 1993).
In equation (4.36) the resonant frequency of the transponder is expressed as the
product L
2
C
2
. This is not recognisable at first glance. In order to make a direct pre-
diction regarding the frequency dependency of interrogation sensitivity, we rearrange
equation (4.25) to obtain:
L
2
C
2
=
1
(2πf
0
)
2
=

1
ω
2
0
(4.37)
By substituting this expression into the right-hand term under the root of
equation (4.36) we obtain a function in which the dependence of the interrogation field
strength H
min
on the relationship between the transmission frequency of the reader (ω)
and the resonant frequency of the transponder (ω
0
) is clearly expressed. This is based
upon the assumption that the change in the resonant frequency of the transponder is
caused by a change in the capacitance of C
2
(e.g. due to temperature dependence or
manufacturing tolerances of this capacitance), whereas the inductance L
2
of the coil
remains constant. To express this, the capacitor C
2
in the left-hand term under the root
of equation (4.36) is replaced by C
2
= (ω
2
0
· L
2

)
−1
:
H
min
=
u
2
·

ω
2

L
2
R
L
+
R
2
ω
2
0
L
2

2
+

ω

2
0
− ω
2
ω
2
0
+
R
2
R
L

2
ωµ
0
· A ·N
(4.38)
Therefore a deviation of the transponder resonant frequency from the transmission
frequency of the reader will lead to a higher transponder interrogation field strength
and thus to a lower read range (Figure 4.21).
4.1.9.1 Energy range of transponder systems
If the interrogation field strength of a transponder is known, then we can also assess
the energy range associated with a certain reader. The energy range of a transponder is
the distance from the reader antenna at which there is just enough energy to operate the
transponder (defined by u
2min
and R
L
). However, the question of whether the energy

range obtained corresponds with the maximum functional range of the system also
depends upon whether the data transmitted from the transponder can be detected by
the reader at the distance in question.
Given a known antenna current
2
I ,radiusR, and number of windings of the trans-
mitter antenna N
1
, the path of the field strength in the x direction can be calculated
using equation (4.3) (see Section 4.1.1.1). If we solve the equation with respect to x
2
If the antenna current of the transmitter antenna is not known it can be calculated from the measured field
strength H(x) at a distance x, where the antenna radius R and the number of windings N
1
are known (see
Section 4.1.1.1).
4.1 MAGNETIC FIELD 83
0.5
1
1.5
2
2.5
3
1 × 10
7
Resonant frequency (MHz)
R
L
= 1500 Ohm
Interrogation field strength

H
min
1.4 × 10
7
1.2 × 10
7
1.6 × 10
7
1.8 × 10
7
2 × 10
7
Interrogation field strength (A/m)
Figure 4.21 Interrogation sensitivity of a contactless smart card where the transponder res-
onant frequency is detuned in the range 10–20 MHz (N = 4, A = 0.05 ×0.08 m
2
, u
2
= 5V,
L
2
= 3.5 µH, R
2
= 5 , R
L
= 1.5k). If the transponder resonant frequency deviates from the
transmission frequency (13.56 MHz) of the reader an increasingly high field strength is required
to address the transponder. In practical operation this results in a reduction of the read range
we obtain the following relationship between the energy range and interrogation field
H

min
of a transponder for a given reader:
x =




3


I · N
1
· R
2
2 ·H
min

2
− R
2
(4.39)
As an example (see Figure 4.22), let us now consider the energy range of a transpon-
der as a function of the power consumption of the data carrier (R
L
= u
2
/i
2
). The reader
in this example generates a field strength of 0.115 A/m at a distance of 80 cm from the

transmitter antenna (radius R of transmitter antenna: 40 cm). This is a typical value
for RFID systems in accordance with ISO 15693.
As the current consumption of the transponder (lower R
L
) increases, the interroga-
tion sensitivity of the transponder also increases and the energy range falls.
The maximum energy range of the transponder is determined by the distance
between transponder and reader antenna at which the minimum power supply u
2min
required for the operation of the data carrier exists even with an unloaded transponder
resonant circuit (i.e. i
2
→ 0, R
L
→∞). Where distance x = 0 the maximum current
i
2
represents a limit, above which the supply voltage for the data carrier falls below
u
2min
, which means that the reliable operation of the data carrier can no longer be
guaranteed in this operating state.
84 4 PHYSICAL PRINCIPLES OF RFID SYSTEMS
0.5
Energy range (m)
1
1.5
Vcc = 5 V
1 × 10
−6

1 × 10
−5
1 × 10
−3
1 × 10
−2
0.01
Power consumption of data carrier (A)
Figure 4.22 The energy range of a transponder also depends upon the power consumption of
the data carrier (R
L
). The transmitter antenna of the simulated system generates a field strength
of 0.115 A/m at a distance of 80 cm, a value typical for RFID systems in accordance with ISO
15693 (transmitter: I = 1A, N
1
= 1, R = 0.4 m. Transponder: A = 0.048 × 0.076 m
2
(smart
card), N = 4, L
2
= 3.6 µH, u
2min
= 5V/3V)
4.1.9.2 Interrogation zone of readers
In the calculations above the implicit assumption was made of a homogeneous magnetic
field H parallel to the coil axis x. A glance at Figure 4.23 shows that this only applies
for an arrangement of reader coil and transponder coil with a common central axis x.
If the transponder is tilted away from this central axis or displaced in the direction of
the y or z axis this condition is no longer fulfilled.
If a coil is magnetised by a magnetic field H , which is tilted by the angle ϑ in

relation to the central axis of the coil, then in very general terms the following applies:
u
0ϑ
= u
0
· cos(ϑ) (4.40)
where u
0
is the voltage that is induced when the coil is perpendicular to the magnetic
field. At an angle ϑ = 90
◦
, in which case the field lines run in the plane of the coil
radius R, no voltage is induced in the coil.
As a result of the bending of the magnetic field lines in the entire area around the
reader coil, here too there are different angles ϑ of the magnetic field H in relation
to the transponder coil. This leads to a characteristic interrogation zone (Figure 4.24,
grey area) around the reader antenna. Areas with an angle ϑ = 0
◦
in relation to the
transponder antenna — for example along the coil axis x, but also to the side of the
antenna windings (returning field lines) — give rise to an optimal read range. Areas in
which the magnetic field lines run parallel to the plane of the transponder coil radius
4.1 MAGNETIC FIELD 85
2
R
2
R
T
x
Reader antenna

Transponder antenna
Figure 4.23 Cross-section through reader and transponder antennas. The transponder antenna
is tilted at an angle ϑ in relation to the reader antenna
Magnetic field line
H
Transponder coil
Transponder coil
cross-section
Reading range with
parallel transponder coil
Reading range with
vertical transponder coil
Reader antenna
(cross-section)
Figure 4.24 Interrogation zone of a reader at different alignments of the transponder coil
R — for example, exactly above and below the coil windings — exhibit a significantly
reduced read range. If the transponder itself is tilted through 90
◦
a completely different
picture of the interrogation zone emerges (Figure 4.24, dotted line). Field lines that
run parallel to the R-plane of the reader coil now penetrate the transponder coil at an
angle ϑ = 0
◦
and thus lead to an optimal range in this area.